The Ciarlet definition of a finite element has been core to our understanding of the finite element method since its inception. It has proved particularly useful in structuring the implementation of finite element software. However, the definition does not encapsulate all the details required to uniquely implement an element, meaning each user of the definition (whether a researcher or software package) must make further mathematical assumptions to produce a working system. 

The talk presents a new definition built on Ciarlet’s that addresses these concerns. The novel definition forms the core of a new piece of software in development, FUSE, which allows the users to consider the choice of finite element as part of the data they are working with. This is a new implementation strategy among finite element software packages, and we will discuss some potential benefits of the development.

A number of algorithms are now available---including Halko-Martinsson-Tropp, interpolative decomposition, CUR, generalized Nystrom, and QR with column pivoting---for computing a low-rank approximation of matrices. Some methods come with extremely strong guarantees, while others may fail with nonnegligible probability. We present methods for efficiently estimating the error of the approximation for a specific instantiation of the methods. Such certificate allows us to execute "responsibly reckless" algorithms, wherein one tries a fast, but potentially unstable, algorithm, to obtain a potential solution; the quality of the solution is then assessed in a reliable fashion, and remedied if necessary. This is joint work with Gunnar Martinsson. 

Time permitting, I will ramble about other topics in Randomised NLA. 

We’ll begin by introducing homotopy algebras (assuming no background) and their intimate connection to quantum field theory, with a briefly summary of some applications: scattering amplitude recursion relations, colour-kinematics duality, and generalised asymptotic observables. We’ll then introduce (deformed) Alexandrov–Kontsevich–Schwarz–Zaboronsky theories as the paradigmatic example of this framework, before developing their applications to gravity in two, three and four dimensions.   

I am interested in studying moduli spaces and associated enumerative invariants via degeneration techniques. Logarithmic geometry is a natural language for constructing and studying relevant moduli spaces. In this talk I  will explain the logarithmic Hilbert (or more generally Quot) scheme and outline how the construction helps study enumerative invariants associated to Hilbert/Quot schemes- a story we now understand well. Time permitting, I will discuss some challenges and key insights for studying moduli of stable vector bundles/ sheaves via similar techniques - a theory whose details are still being worked out.